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After reading Section III.11 (The theorem on Formal functions) and III.12 (The semicontinuity Theorem), I feel that I get some kind of formalism instead of a clear picture of what is going on. So the big question is how one should understand these two sections. What is the morale behind all the technical endeavor? I especially want to compare this with the section "proper base change theorem" in Milne's online note on étale cohomology (I.17).

In the étale case, if $X\to Y$ is proper, $\mathcal{F}$ is a torsion sheaf on the site $X_{ét}$, and the following diagram is Cartesian, then $u^*R^if_* \mathcal{F}\cong R^ig_*(v^* \mathcal{F})$. $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{ll} X' & \ra{v} & X \\ \da{g} & & \da{f} \\ Y' & \ra{u} & Y \\ \end{array} $$

But in Hartshorne's case, we are dealing with Zariski topology so things are not so nice. For example, it seems to me that Theorem III.11.1 in Hartshornes says that the base change is "true" if we look at infinitesimal neighborhoods of a point in $Y$. I hope someone could make some more remarks (Taking completion is like passing to the infinitesimal neighborhoods, that's pretty much all I understand here. )

For all that follows, let's assume that $Y=\mathrm{Spec}(A)$, and $Y'=\mathrm{Spec}(B)$, and $\mathcal{F}$ a quasicoherent sheaf on $X$. In section III.12, Hartshorne defined a functor $T^i(M)=H^i(X, \mathcal{F}\otimes_A M)$ for any $A$-module $M$. I don't think I quite understand this functor. I see that $T^i(A)\otimes B$ is the module associated with the sheaf $u^*R^if_*\mathcal{F}$. If $B=k(y)$, Corollary 9.4 tells me that $T^i(k(y))$ is the module for $R^i g_*(v^*\mathcal{F})$. But for a general A-algebra $B$, is there any meaning to $T^i(B)$? It seems to be a bit odd that a lot of the work is done for a general $M$, then in the end, all the major theorems restrict to the case $M=k(y)$. For example, in Theorem 12.11, can one go beyond just the fibers and write down some statement like the étale case (with more conditions, certainly)?

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My understanding is that this base-change business works on schemes (maybe on ringed spaces of finite dimension?) if one replaces the pull-back with the derived pull-back. In the etale version of base-change, one is just working with sheaves of abelian groups and so pull-back is automatically exact. – Akhil Mathew Jun 24 '11 at 13:24
Thanks, Akhil. I understand the part of the remark on étale version. But could you elaborate on the first part? I guess that you are talking about some derived category generalization, but my knowledge of derived category is a bit to be desired. – Jiangwei Xue Jun 24 '11 at 14:16
Yeah. My understanding was that, in the above diagram, then $Lu^* Rf_* \simeq Rg_* Lv^*$--granted, one has to be careful on which derived category one works (bounded-above should be OK if, say, one works with noetherian schemes so $Rg_*$ is well-defined). I was told this before I learned what a derived category was, though, so it could be wrong... – Akhil Mathew Jun 24 '11 at 17:24
@Jianwei: Sorry, I'm actually wrong. Even for quasi-coherent sheaves, this is apparently false because the scheme $X'$ itself should be taken in the "derived" sense! If one works with derived schemes, then the base-change isomorphism is an isomorphism (even without properness hypotheses). (I just checked this claim with Dennis Gaitsgory and he says there is no source at this point...) – Akhil Mathew Jun 24 '11 at 21:01

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